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Chapter 3
1
Section 3.1 – Investigating
quadratic functions in vertex form
TERMINOLOGY
Parabola: Symmetrical curve
Graph of quadratic function
Vertex: lowest or highest point of
parabola (depends if it opens up or down)
Axis of symmetry: a line that
divides a parabola in half
AOS = x-value of vertex
Minimum and Maximum Values: least or greatest value in range of function
(parabola) .
If Minimum Value given = Graph opens up
If Maximum Value given = Graph opens down
What is a linear function?
What is a quadratic function?
𝑦 = 𝑚𝑥 + 𝑏
𝑜𝑟
𝑓(𝑥) = 𝑚𝑥 + 𝑏
𝑦 = 𝑚𝑥 2 + 𝑏
or
𝑓(𝑥) = 𝑚𝑥 2 + 𝑏
Vertex Form f(x) = 𝑎(𝑥 − 𝑝)2 + 𝑞 where a, p, q are constants and a ≠ 0
Vertex will be (p,q)
Axis of symmetry = p or x=p
If a > 0 graph opens up = minimum value
If a < 0 graph opens down = maximum value
2
Lets Explore!!!
Graph and Sketch
f(x) = 𝑥 2
f(x) = 4𝑥 2 ….
1
f(x) = 4 𝑥 2 ….
f(x) = −4𝑥 2
1
f(x) = − 4 𝑥 2
How does it compare to f(x) = 𝒙𝟐
a>0
a<0
-1 < a < 1
a > 1 or a < -1
Graph and Sketch
f(x) = 𝑥 2
f(x) = 𝑥 2 + 2
f(x) = 𝑥 2 − 2
How does it compare to f(x) = 𝒙𝟐
f(x) = 𝑥 2 + q
f(x) = 𝑥 2 - q
Graph and Sketch
f(x) = 𝑥 2
f(x) = (𝑥 − 3)2
f(x) = (𝑥 + 3)2
f(x) = (𝑥 − 𝑝)2
f(x) = (𝑥 + 𝑝)2
3
So, what does "p" do?
But why does p appear to work backwards? The vertex form of a quadratic is y  a  x  p   q .
2
Notice how there is a subtraction in front of the p? Notice how there is a subtraction in front of
the p? So, consider, y   x  2  . The p value is actually 2! So, a positive 2 means 2 unit RIGHT.
2


Now consider y   x  4  . That is equivalent to y  x   4 . Thus, p is 4 , so four units left!
2
2
If there is more than one transformation…Do any reflections
or stretches first AND then translations!!!
Example 1: Determine the following characteristics for the function
y=2(𝑥 − 4)2 + 1
a) Vertex b) domain and range c) direction of opening d) equation of axis of symmetry
e) Create a sketch (two methods…1: Transformations 2: using points and symmetry)!!
Example 2: Determine the characteristics as in “example 1” for the equation y = -3(x–2)2 + 5
4
YOUR TURN
Hints:
Vertex
equation of axis of symmetry
domain & range
*
y-intercept
2nd Calc Value x  0
x-intercept(s) 
graphing calculator for most questions
2nd calc zero
find x-intercepts without calculator.
** find x-intercepts with calculator.
Vertex
*
2nd calc max or min
x = value from vertex
always x  R , y  or y  value from vertex
Axis of
Sym.
Direction
Of opening
(up or down)
Width
Compare
to
y  x2
(narrower,
wider or
the same)
Domain
and
Range
Max or
Min
Value?
What is
it?
y-int
1. y = (x–2)2 + 5
2. y = 3x2 – 12
3. y = 2x2  4
*
4. y = (x  2)2
*
1
5. y =  (x + 3)2
2
**6.
y = 2(x – 4)2  3
7. y = 6(x + 3)2 + m
8. y = √3(𝑥 − 1)2 + √3
5
x-int
Vertex Form …………y= 𝑎(𝑥 − 𝑝)2 + 𝑞
Example 1: Determine a quadratic function in vertex form for information given (examples 1-3).
a = 5 and vertex is (4, 1)
Example 2: congruent to y = 3x2, opens down, and vertex is (4, 9)
Rope Swing! Vertex is
Y
10
9
8
7
6
Example
5
4
3
2
1
4: Determine a quadratic function in vertex form for each graph.
X
0
6 -5 -4 -3 -2 -1
1 2 3 4 5 6 7 8 9 10
-1
Y
-2 10
-3 9
-4 8
-5
7
-6
6
-7
Example
5: Determine a quadratic function
-8 5
-9 4
-10 3
2
1
in vertex form for each graph.
unregistered version of Advanced Grapher - http:/ / www.serpik.com/ agrapher/
-8 -7 -6 -5 -4 -3 -2 -1-10
-2
-3
-4
-5
-6
-7
-8
-9
-10
X
1 2 3 4 5 6 7 8 9 10
d with an unregistered version of Advanced Grapher - http:/ / www.serpik.com/ agrapher/
6
Example 6: Determine the number of x-intercepts for each quadratic function without graphing
a) f(x) = 0.5𝑥 2 − 7
1
b) f(x) = −2(𝑥 + 1)2
c) f(x) =− 6 (𝑥 − 5)2 − 11
To figure out x-intercept
1) substitute 0 in for y
or
2) 2nd trace ZERO (Left side of intersection point and right side of intersection point)
Questions:
Vertex
Axis of
Symmetry
1.
Range
y-int
x-int
Does the graph
Width
have a max or min
compare
& value? What is
d to y = x2
it?
y = (x – 6)2 + 3
y=
1
(x  2)2 + 7
2
f  x   3  x  1  4
2
f  x
1
(x  6)2  8
2
f  x   4 x2 12 x  9
calculator only
y  0.6 x 2  5x  13
calculator only
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2. congruent to y = 6x2, range is y  3, and axis of symmetry is x = 5
3. A parabolic vase is 20 cm high and 4 cm wide. Find an equation to represent this vase.
4. A parabolic archway is 12 m wide and 9 m high as shown below. Find an equation to
represent this archway.
(0, 9)
Y
(6, 0)
10
9
5. Write an8 equation for the parabola: vertex is (0,
7
6
6. Write an5 equation for each parabola:
4
3
2
1
3
) passing thru (3, 3)
2
10
9
8
7
6
5
4
3
2
1
X
10 -9 -8 -7 -6 -5 -4 -3 -2 -1-10 1 2 3 4 5 6 7 8 9 10
-2
-3
-4
-5
-6
-7
-8
-9
7. Two points
-10 on a parabola are (5, 2) and (9, 2).
Y
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1-10 1 2 3 4 5 6 7 8 9 10
-2
-3
-4
What is the equation of the axis
-5 of
symmetry? (Can not be done with equation. Must use logic and symmetry.) -6
Created with an unregistered version of Advanced Grapher - http:/ / www.serpik.com/ agrapher/
-7
8. Write an equation for the parabola with a minimum at (3, 2), passes thru (5,-86)
-9
-10
9. The vertex is (3, 12), x-intercept is 5, what is the other x-intercept?
Created with an unregistered version of Advanced Grapher - http:/ / www.se
HOMEWORK Page (1,2 a,c only),
3c,
4a, d
7,
8,
10, 13a, 14,
21
8
Section 3.2 – Investigating Quadratic Functions in
Standard Form
Vertex Form
y= 𝑎(𝑥 − 𝑝)2 + 𝑞
Standard Form
y= ax2+ bx + c
where a,b,c are real numbers AND
a≠0
In Standard Form:



“a” determines the shape (width of graph…a small number is wider, where a larger
number is narrower) and whether the graph opens upward (positive “a”) or downward
(negative “a”)
“b” influences the position of the graph
“c” is the y-intercept of the graph
HOW DO WE GO FROM VERTEX FORM TO STANDARD FORM…..EXPAND!!!
Example 1: Convert y = 2(x  3)2  11 into standard form
Example 2: Convert the following equation from vertex form to standard form
f(x) = 𝑎(𝑥 − 𝑝)2 + 𝑞
Therefore we also know that the x- coordinate of the vertex (or axis of symmetry) is
x= -
𝒃
𝟐𝒂
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Example 3: What is the vertex of y  3x 2  9 x  1?
Example 4: Use the equation and the graph to answer the questions below.
f  x   x2  8x  20
a) the vertex b) equation of the axis of symmetry c) x and y – intercepts d) domain and range
Example 5: The path of a basketball shot is modelled by h(t) = 1.2t2 + 3.3t + 2.2 where
h(t) = height of ball in m, and t = time in seconds. Find:  0.01 Nearest hundredth when:
a.
Maximum height reached
b.
Time when max is reached
c.
What is the y-intercept and what does it mean?
d.
Time when the ball hits the floor.
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Questions:
For questions 1-3, identify the following:
a) direction of opening b) vertex
c) max or min value d) axis of symmetry
e) x & y intercepts
f) domain and range
1) f(x) = −𝑥 2 + 2𝑥 + 3
2) f(x) = 𝑥 2 + 6𝑥 + 5
3) f(x) = 𝑥 2 + 6𝑥
4) Mr. H performs a triple sow cow double flapjack underside up somersault jump from a 3-m
springboard with an initial vertical velocity of 6.8 m/s. His height, h, in metres, above the water t
seconds after leaving the diving board can be modelled by the function h(t) = −4.9𝑡 2 + 6.8𝑡 + 3
a) graph the function
b) what does the y-intercept represent
c) when and what is the max height reached
d) how long does it take before the diver hits the water
e) domain and range
f) what is the height of the diver 0.6 seconds after leaving the board
HOMEWORK Questions PAGE 174 # 1,
2a,
3,
5a,d ,
6a,b
8, 10a, d,
12, 14a, 16
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SECTION 3.3 – COMPLETING THE SQUARE
VERTEX  Standard
From Section 3.2…we looked at converting vertex form to standard form
Example 1: Convert to standard form:
𝑦 + 3 = 2(𝑥 + 3)2
Example 2: Convert to standard form: 𝑦 − 2 = 3(𝑥 − 2)2
Standard  VERTEX
NOW LET’S DO A REVERSAL LET’S GO FROM STANDARD BACK TO VERTEX FORM
Example 3: Convert x2 + 8x – 7 = y into vertex form
Step 1: Group terms with x, and take other constants (numbers) or terms (like y) to the other side
Step 2: Put brackets around the x-terms and leave a space at the end of the bracket
Step 3: Make sure the x2 term has a coefficient of 1. If a coefficient is found in front (not 1)
factor it out!
Step 4: Divide the x-coefficient by 2, square the result, and add inside the bracket.
Step 5: Multiply this number with the coefficient in front of the brackets. Take its opposite and
add it to the end of the question.
Step 6: Factor the expressions
Step 7: Rewrite equation so it is in vertex form
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Example 4: Convert -3x2 – 18x – 24=y into vertex form
Example 5: Convert 4x2 -28x -23 = y into vertex form
Questions
Convert the following questions from standard form into vertex form
1) y = 2x2  12x + 7
2) y = 3x2 + 12x  15
3) y = x2 + 10x  3
4) y + 5x2 + 8x = 0
5) y =
2 2 3
x  x2
3
5
6) y = 30x2 + 1200x + 1600
7) y = 3x2  30x
8) y = 2x2  20x + p
9) y=
5x 2  4 10 x  2 5 HARD!!!
HOMEWORK Page 192 2 a, d 3 d 4 c 5 a 6 d 7 c, f 12 ac (find error only) 13, 15a 16a
13
Problem solving day!!!
Function is Given
Example 1. The path of a basketball shot is modelled by h(t) = t2 + 3t + 2.2 where h(t) = height
of ball in m, and t = time in seconds. Find:
a. The maximum height reached
b. The time when max is reached
c. The distance ball is from floor when player releases it
d. time when the ball hits the floor.
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B. Area Problems
Example 2: Farmer John has 240 m of fence. Find dimensions so that area is a maximum.
y
x
2.
Another farmer John has 80 m of fence. Find dimensions so that area is a maximum.
Existing fence
x
y
15
3. A third farmer John has 200 m of fence. Find dimensions so that area is a maximum.
y
X
m2
Extra Question
A fourth farm John has 300 m of fence. Find dimensions so that area is a maximum.
X
y
16
C. Income Problems
Examples: 1. You sell 60 books if the price is $8 or less. For every $2 increase, sales drop by 6.
Find: a. the maximum income
B. the price and number of books to yield the maximum income
Let x = number of increases
x
# of books
price/book
0
60
$8
1
2
54
48
(60  6)
$10
(8 + 2(1))
(60  2(6))
$12
(8 + 2(2))
3
.
.
.
.
.
.
x
60  6x
8 + 2x
2. 400 articles bought if price is $5.00. Each increase of $0.05 results in 2 fewer sales. Find
maximum income and price and # of articles that yield this max.
17
3. A riverboat cruise ride is $36/person. Cruise averages 300 people a day. Each $2
increase results in 10 fewer customers. What increase in price would yield maximum income?
4. Two numbers sum is 56. What is their product if it is a maximum?
Assignment
Page 194 14
General Word Problem
17
General Word Problem (small decimals)
18
Income Problem
19
Income Problem
31
Income Problem
24
Area Problem
Page 177 17
Area Problem
Page 196 23
problems.
Solve Algebraically
Two Number problems, not taught in lessons but similar to area
18
Review!!!!!!!!!
1. A quadratic function has its vertex at (3, 4). If one point of the function is (7, 20), find the
other point.
2.
The vertex of a quadratic function is (3, 8). One x-intercept is 4. What is the other
one?
3.
The coordinates of the vertex of y = x2 – 8x + 2 are:
A. (–4, –14)
4.
B. (–4, 2)
C. (4, –14)
D. (4, 2)
Sketch y  2 x 2  8 x  1 on the grid below. You must show the vertex and at least 2
other points.
5.
A cannon ball is fired from a cannon and follows a path given by h = –0.2t2 + 4t + 2.3
where h is height in metres and t is time in seconds.
What is the maximum height? Or, when does the maximum height occur?
19